What does the discriminant tell you?

For ax² + bx + c = 0 the discriminant is b² - 4ac. Positive means two distinct real roots, zero means two equal roots, and negative means no real roots.

How do I find when a line is a tangent to a curve?

Substitute the line into the curve to get one quadratic in x, then set its discriminant equal to zero. A tangent meets the curve at exactly one point.

Why is my range of k the wrong way round?

A quadratic inequality in k has its solution outside the roots when the inequality is > 0 and between them when it is < 0. Sketch the parabola in k to see which.

4037 Discriminant Questions: Nature of Roots, Tangents

On 4037 the discriminant is rarely asked about on its own. It comes wrapped in a condition: equal roots, no real roots, or a line that touches or crosses a curve. Each question reduces to the same move: get one quadratic, write $b^{2} - 4 a c$ , and apply the right inequality.

Question 1: equal roots [2 marks]

Find the values of $k$ for which $x^{2} + k x + 9 = 0$ has two equal roots.

Show the worked solution

Equal roots means $b^{2} - 4 a c = 0$ , with $a = 1$ , $b = k$ and $c = 9$ :

k^{2} - 36 = 0 ⟹ k = 6 or k = - 6

Both values count. Giving only $k = 6$ loses a mark.

Question 2: no real roots [2 marks]

Find the range of values of $k$ for which $2 x^{2} + 4 x + k = 0$ has no real roots.

Show the worked solution

No real roots means $b^{2} - 4 a c < 0$ :

16 - 8 k < 0 ⟹ 8 k > 16 ⟹ k > 2

Question 3: a tangent line [4 marks]

Find the values of $k$ for which the line $y = k x + 1$ is a tangent to the curve $y = x^{2} - 3 x + 5$ .

Show the worked solution

At a meeting point the two $y$ values are equal:

x^{2} - 3 x + 5 = k x + 1 ⟹ x^{2} - (3 + k) x + 4 = 0

A tangent meets the curve once, so the discriminant is zero:

(3 + k)^{2} - 16 = 0 ⟹ 3 + k = \pm 4 ⟹ k = 1 or k = - 7

Check $k = 1$ : the equation becomes $x^{2} - 4 x + 4 = (x - 2)^{2} = 0$ , one repeated root, so the line touches at $(2, 3)$ .

Question 4: two distinct points [3 marks]

Using the same line and curve, find the range of values of $k$ for which the line meets the curve at two distinct points.

Show the worked solution

Two distinct points means the discriminant is positive:

(3 + k)^{2} - 16 > 0

The critical values are $k = 1$ and $k = - 7$ from Question 3. This parabola in $k$ opens upwards, so it is positive outside its roots:

k < - 7 or k > 1

Question 5: show real roots for every k [3 marks]

Show that $x^{2} + (k - 2) x - k = 0$ has two distinct real roots for all real values of $k$ .

Show the worked solution

b^{2} - 4 a c = (k - 2)^{2} - 4 (1) (- k) = k^{2} - 4 k + 4 + 4 k = k^{2} + 4

$k^{2} \geq 0$ for every real $k$ , so $k^{2} + 4 \geq 4 > 0$ . The discriminant is always positive, so there are always two distinct real roots.

Try it: slide k and watch the roots

Place the critical values of $k$ first, then slide $k$ to see the roots cross, touch and vanish.

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O Level Additional Maths 4037: discriminant and nature of roots questions

Question 1: equal roots [2 marks]

Question 2: no real roots [2 marks]

Question 3: a tangent line [4 marks]

Question 4: two distinct points [3 marks]

Question 5: show real roots for every k [3 marks]

Try it: slide k and watch the roots

Questions students ask